Sigma notation abstract limits What is meant by the following  notations? $\sum\limits_{k=0}^{p+1}k^3$, $\sum\limits_{k=0}^{p-2}k^3$ I need to use this notation to prove a statement is true for $\sum\limits_{k=0}^{p+1}k^3$ when $\sum\limits_{k=0}^{p}k^3$ is already proved, and other situations alike.
Is there a consistent documentation of the notation used in mathematics, which people have agreed on? Either a book or an online resource recommendation would be great.
 A: Yes, the notation is consistent. The following: $$S=\sum_{k=n_1}^{n_2} f\left(k\right)$$ means precisely: $$S= f\left(n_1\right) +  f\left(n_1+1\right) + \cdots +  f\left(n_2-1\right) +  f\left(n_2\right)$$
In your case, we have:
$$\sum_{k=0}^{p+1} k^3 = 0^3 + 1^3 + \cdots + p^3 + \left(p+1\right)^3$$
You might want to use the following property when trying to prove what you want:
$$\sum_{k=0}^{p+1}k^3 =\sum_{k=0}^{p}k^3 + \left(p+1\right)^3 $$
which means that:
$$\left[0^3 + 1^3 + \cdots + p^3\right] + \left(p+1\right)^3 =
\left[0^3 + 1^3 + \cdots + p^3 + \left(p+1\right)^3\right]$$

I should also mention that sometimes, when the set of values that we want $k$ to range over is not of the form $\left\{n_1,n_1+1,\ldots,n_2\right\}$, we use the following notation: $$S= \sum_{k\in A} f\left(k\right)$$ to mean that we want the sum for all $k$ in the set $A$.
For instance, if $A$ is the set of prime numbers less than $10$, $A=\left\{2,3,5,7\right\}$, we will write:
$$\sum_{k\in A} k^3 = 2^3 + 3^3 + 5^3 + 7^3$$
